## Fourier analysis on matrix space |

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ad-bc analogue analytic continuation bx+d bz+d Chapter compactly supported completes the proof conjugacy class constant on cosets continuous series cosets denote the representation det(x diagonal subgroup disconnected discrete series eigenvalues elements factor feC G fF(h ffC G ffS(k follows Fourier operator Fourier transform function f function on G functional equation GL(n Haar measure initially defined integral defining integrals converge inversion method Iwasawa decomposition Lemma let X denote m m m matrix Mellin transform modulo P(n multiplicative character n,t n,t Plancherel formula Plancherel measure Plancherel theorem principal series Problem proof of Lemma proof of Theorem Proposition prove Theorem representation of G representation space representation theory resp series for G singular matrix subgroup H summable Suppose Tate Tate's theorem for G unimodular upper triangular subgroup Weyl group zeta-function